Advances in Water Resources

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1 Advances in Water Resources 34 (2) Contents lists available at ScienceDirect Advances in Water Resources journal homepage: Unsteady overland low on lat suraces induced by spatial permeability contrasts Sally Thompson a,, Gabriel Katul a,b, Alexandra Konings a, Luca Ridoli b a Nicholas School o the Environment, Duke University, Durham, NC 2777, USA b Dipartimento di Idraulica, Trasporti e Inrastrutture Civili, Politecnico di Torino, C.so Duca degli Abruzzi, 24, 29 Torino, Italy article ino abstract Article history: Received 9 February 2 Received in revised orm 3 May 2 Accepted 3 May 2 Available online 2 June 2 Keywords: Arid ecosystems Hydraulics Iniltration Manning s equation Overland low Patchy vegetation Lateral redistribution o surace water in patchy arid ecosystems has been hypothesized to contribute to the maintenance o vegetation patches through the provision o a water subsidy rom bare sites to vegetated sites. Such runon runo processes occur during Hortonian runo events on topographically sloping ground. Surace low redistribution may also occur on topographically lat ground i the presence o the vegetation patch creates a contrast in iniltration rate, leading to a ree-surace gradient in ponded water. The precise dynamics and the eco-hydrologic role o this process has resisted complete theoretical treatment to date. Here the overland low equations are modiied to account or the presence o vegetation situated over a lat surace. The resulting model is solved numerically to determine whether this mechanism could inluence the spatial partitioning o water in patchy arid ecosystems. Assumptions made about iniltration processes and overland low in existing eco-hydrologic models o patchy and patterned arid ecosystems are evaluated in comparison to the solution o the ull coupled Saint Venant equations with various iniltration models. The results indicate that the optimization o vegetation spatial patch scales with respect to water redistribution may be determined by the size o the iniltration redistribution length L over which the presence o an iniltration contrast perturbs baseline iniltration behavior. Ó 2 Elsevier Ltd. All rights reserved.. Introduction The lateral redistribution o ponded water generated during intense rainstorms plays an important role in the hydrological, biogeochemical and ecological unctioning o patchy arid ecosystems. For instance, it is broadly accepted that the maintenance o vegetation in patchy arid systems is supported by lateral subsidies o rainall rom bare sites to vegetated ones [ 4]. However, the relative importance o above ground and below ground (via root uptake) water redistribution processes remains unresolved, particularly on lat sites [5 8]. At those lat sites, one working hypothesis is that the combined iniltration contrast and the eiciency o surace lateral transport alone can generate suicient subsidies o water rom bare sites to satisy water demand in vegetated patches [6 8]. The potential energy gradients that drive overland low can arise either rom the topographic slope o the soil surace, or, in the absence o slope, rom gradients in the ree water surace. The ormer situation is typically assumed in hydrological routing models, and presumes that on lat terrain hydrological routing collapses to a simple one dimensional iniltration problem [9 ]. Redistribution o water in response to a ree-surace gradient in the absence o a topographic slope, however, is known to be important in engineering applications such as irrigation or dam-removal Corresponding author. address: set8@duke.edu (S. Thompson). [2,3]. On lat ground, which occurs in several areas with vegetation patterning [8], similar gradients could hypothetically arise during Hortonian runo generation i a pronounced spatial contrast in rainall intensity or iniltration rate create contrasting depths in ponded head through space. These mechanisms persist in the presence o a ground slope but become less important or sustaining lateral low as the ground slope becomes large compared to the ree surace gradients. The pronounced contrasts in iniltration rate needed to generate this ree-surace contrast can arise in patchily-vegetated arid ecosystems. In many such ecosystems, dierent soil surace conditions are ound in vegetated and bare sites. These dierences can result in -old dierences in iniltration capacity over relatively short distances (a ew meters) [4,5]. Both physical and biological processes are involved in sustaining these dierences. Enhanced biological activity in vegetated sites leads to increased soil macroporosity, oten in association with termite activity [6 8]. The presence o the plant canopy and litter layer protect the soil surace rom rain splash impacts and prevent the ormation o photosynthetic biological soil crusts [9]. By comparison, bare sites are requently colonized by crust-orming biota, and orm physical soil crusts and seals in response to rain splash impact [2 22]. Although individual plants may be associated with shallow mound ormation, these mounds are also associated with enhanced iniltration capacities. The proposed redistribution mechanism can still apply, although the routing would need to be amended to account or the microtopographic geometry [23] /$ - see ront matter Ó 2 Elsevier Ltd. All rights reserved. doi:.6/j.advwatres.2.5.2

2 5 S. Thompson et al. / Advances in Water Resources 34 (2) The lateral luxes o water that arise rom the contrasts in ponded depth are a unction o the water depth h, its spatial and the hydrodynamic roughness o the ground surace. In patchily vegetated sites, the spatial contrasts in vegetation also lead to a contrast in hydrodynamic roughness, which tends to oppose the low [24]. Thus the lateral redistribution o rainall rom bare sites to vegetated sites arises rom a dynamic interplay o hydrological (iniltration contrasts) and hydraulic (roughness contrasts) processes. The spatial dynamics o vegetation and their link to water availability in patchy ecosystems with low slopes have been addressed theoretically by several investigators using partial-dierentialequation based models o coupled biomass-water movement [6,8,25,26]. Such simple models reproduce the observed morphology o patchy vegetation in arid ecosystems. To date, no model has explicitly conronted a mechanistic description o the unsteady hydraulic and hydrological aspects o the low problems, but have instead made two simpliying assumptions:. The iniltration rate ( c ) could be represented as being linearly dependent upon the depth o ponded water (h); and 2. The overland low (q) in an arbitrary direction x linearly scales The irst o these assumptions, although non-standard compared to conventional iniltration theory, gains support rom the role o microtopographic variability and macroporosity in iniltration dynamics. For instance, Dunne et al. [27] showed that where point-scale iniltration capacity was correlated with elevation along microtopographic proiles, bulk iniltration rates depended on the depth o inundation. Fox et al. [28] demonstrated an apparent depth dependence on iniltration capacity arising due to similar mechanisms in crusted soils. Novak et al. [29] similarly showed that the eects o crack structures on iniltration dynamics could be parameterized by treating the iniltration lux as depth-dependent. Both microtopographic variation and macroporosity are likely to arise in vegetated sites, and this h-dependent iniltration representation may be reasonable in a irst-order analysis. The second assumption, however, removes the low-depth nonlinearities arising rom the shallow-water equations. Its use is primarily based on mathematical tractability rather than physical reasoning. Existing theoretical models suggest that redistribution o water in response to the iniltration contrast on lat terrain may be suicient to sustain the patchy characteristics o arid ecosystems [6,8]. However, it is unclear whether these results are biased by the assumptions made about iniltration and hydraulics described above. The goal o this work is to assess whether the lateral redistribution o water induced by the iniltration contrast can generate suiciently large luxes o water to meet water demand in these systems. The study is acilitated by two recent theoretical developments. The irst is a novel representation o low resistance associated with vegetation canopy density (presented below), and the second is a meta-analysis linking above-ground vegetation biomass to the enhancement o iniltration capacity in xeric sites [4]. These results are incorporated into a parameterization o the Saint Venant equations, and are used to explore the redistribution o water rom bare to vegetated sites as driven by variable iniltration rates. This exploration is organized around two key research questions:. Can contrasts in iniltration rate and surace roughness lead to a lateral redistribution o water rom bare to vegetated sites? How large is this redistribution, over what spatial scales does it act, and how sensitive is it to soil, vegetation and storm properties? and 2. How sensitive are the predictions about the lateral redistribution to the assumptions made about iniltration, roughness, and to the choice o hydraulic model used to parameterize the low? The theoretical description o the problem is developed irst, ollowed by a report on numerical investigations aimed at addressing these research questions. 2. Theory The basic low equations describing this problem are the Saint Venant equations, modiied to account or the inluence o vegetation. The Saint Venant equations are widely used to describe shallow overland lows, or which the assumptions that underpin their derivation, namely that the low is nearly one-dimensional and that the momentum equation can be vertically averaged without generating spurious dispersive terms, are generally valid. 2.. The Saint Venant equations The Saint Venant equations describe conservation o mass (continuity, ()) and momentum (2) in a depth-averaged low ¼ cpðtþ @x q 2 x h þ gh2 2! þ ghðs S Þ¼ In these equations, t is time, x is the spatial coordinate, h is the ponded water depth, q x = Vh is the low rate in the x direction, V is the depth averaged velocity, g is gravitational acceleration, P(t) is rainall, I(t) is the iniltration rate, S is the ground slope and S is the riction slope. All symbols used in the text are summarized in Table. We ignore the role o interception in changing rainall luxes at vegetated sites (c = throughout) given that plant canopies in arid systems are generally sparse, displaying unctional adaptations to reduce iniltration losses or to promote the delivery o rainall to the soil surace [3,3]. Parameterizations o the riction slope in the momentum equation (2) are commonly obtained rom the Darcy Weisbach equation assuming the low is locally uniorm, giving: S q 2 x 8g h 3 where is the Darcy Weisbach riction actor. An expression or that accounts or the eects o vegetation on the low hydraulics is developed later Approximations to the Saint Venant equations Several existing ecohydrological models describing water redistribution in arid or semi-arid environments have been proposed that are driven by simpliied versions o the Saint Venant equations [6,8,25,26]. These simpliications are based on retaining the unsteady contribution in the conservation o mass equation but invoking a steady-state assumption or the momentum equation (@q x /@t = ) at small Froude numbers (i.e. q 2 x =h gh2 =2) gh 2 þ ghðs S Þ 2 giving S IS is related to q x via the Darcy Weisbach equation (c.. Eq. 3) then the momentum equation reduces q 2 x 8g h 3 ðþ ð2þ ð3þ ð4þ ð5þ

3 S. Thompson et al. / Advances in Water Resources 34 (2) Table Table o symbols used. Symbol Deinition a Bare soil iniltration rate, cm/h a m Parameter in generalized momentum equation b Momentum absorption coeicient b m Parameter in generalized momentum equation b Exponent o vegetation iniltration relation C d Drag coeicient d Zero plane displacement (m), turbulent Darcy Weisbach riction actor viscous, b, For turbulent, laminar, vegetated or bare soil cases v F, F b, Fv Cumulative iniltration, over bare soil, over vegetation (mm) G(h) Iniltration dependence on ponded depth h Ponded depth (mm) H c Vegetation canopy height (m) I c (t) Iniltration capacity (cm/h) k von Karman s constant k Sensitivity o iniltration capacity to vegetation cover L Iniltration adjustment length (m) l c Adjustment length scale (m) LAI Lea area index M Domain size (m) P(t) Precipitation intensity (cm/h) q x Flow rate in the x direction (m 3 /s) Re Reynold s number S Topographic slope S Friction slope t Time (s) V Bulk (depth averaged) luid velocity (m/s) x Cartesian coordinate aligned along the longitudinal dimension (m) Momentum roughness length (m) z o In almost lat terrain where ; x can be expressed as siiiiii bm q x ¼ h am and the conventional rating curve or low-depth relationship derived solely rom Manning s equation is recovered i a m = 3/2 and b m = /2. Two other simpliications can be used: (i) a m = and b m =, a linear-diusion case where the dependence on h is all lumped into. This dependence is admittedly weak when h is much greater than z o, where z o is the momentum roughness length characterizing the bed surace. The linear diusion assumption recapitulates that used in most existing patchy-vegetation models [6,8,25,26]. (ii) a m = 3/2 and b m =, hereater reerred to as the nonlinear-diusion case because it assumes q x / oh/@x via an h-dependent diusivity. This representation provides a balance between including the ull non-linearities o the shallow water equations and the simpler behavior o the linear diusion case. Note that the assumption x /@t = is questionable under the conditions o abrupt and intense storms, as might be expected in arid environments, motivating the need to undertake a quantitative comparison between the various low routing approaches Surace resistance and the Darcy Weisbach riction actor Determining the most important causes o energetic losses in these shallow low regimes is non-trivial. In examining similar low regimes, Roche et al. [32] concluded that although local Reynolds number values do not clearly indicate that the low is turbulent... head losses... are dominated by inertia [32, p. ]. Rainsplash will tend to agitate the low, contributing to a disturbed state resembling a turbulent regime [33]. Parameterizing rictional losses associated with laminar low in shallow overland low is also problematic, and practical approaches are oten based on extrapolation o Poiseuille s equation or laminar low in pipes, resulting in = 64/Re where Re is the Reynold s Number, Re = hv/m, and m is the kinematic viscosity o water. This extrapolation, however, rarely perorms well in capturing the relationship between Reynold s number, roughness and bulk velocity [34,35] primarily due to the highly disturbed state o the boundary layer over these roughness elements. Given the uncertainty regarding the validity o the Poiseuille approximation or these shallow lows, the majority o the analysis here is based upon the assumption that the turbulent stresses are the dominant contributor to and that laminar stresses can be omitted when the turbulent stresses are considered. That is, we make the reasonable assumption that the turbulent intensity o the shallow lows is high, even when the bulk velocity is low. To explore the implications o this assumption, we also include a treatment in which the turbulent stresses are ignored and only the laminar stresses as given by = 64/Re are included. For the very shallow depths o low that arise in this problem, this approximation is likely to induce such large stresses as to render the water virtually stationary and in many cases reduce the problem to a close approximation o one dimensional iniltration. Thus, a ocus on the turbulent case allows us to explore a limiting scenario with the maximum water redistribution Turbulent low The surace resistance is a nonlinear unction o h, and these unctions dier or the case o low over bare soil versus through emergent vegetation. In the case o bare soil, the depth dependence o the Darcy Weisbach riction actor is the classical /7 power law [36 39]. Note that this is a deep layer ormulation that assumes that a momentum roughness height z o < h: riii :8 z o =7: ð7þ 8 h In the presence o vegetation, two dierent regimes must be speciied or the cases where the low depth h exceeds the vegetation canopy height H c, and where the canopy is emergent, i.e. h/h c <. For the ormer case: riii ¼ 2b l c exp H c þ h 8 H c 2b 2 l c k H c " " h d # d # h d h Hc H c d k Hc þ log ð8þ z o z o while or h/h c <, riii ¼ 2b l c H c h exp þ exp ; ð9þ 8 h 2b 2 l c 2b 2 l c h i qiiii d where ¼ 2b3 lc zo ; ¼ 2b3 lc exp k LAI ; b ¼ min :35 ; :33 Hc khc Hc khc b Hc h i. LAI and l c ¼ C d Hc The ull derivation o these equations is presented elsewhere [4]. The zero-plane displacement d and momentum roughness height z o or the vegetation both vary with the geometric properties o the canopy, while b, the momentum absorption coeicient varies with the lea area density. The adjustment length l c measures how rapidly the wakes generated by the vegetation dissipate turbulent kinetic energy rom the low. The k is Von Karman s constant, and the C d is the dimensionless oliage drag coeicient. In all these ormulations, we assumed that the lea area density was uniorm with height, and was well approximated by the lea area index (LAI) divided by the canopy height (H c ). When the drag orce exerted by the vegetation on the low is dominated by orm rather than viscous-drag, C d is constant. In practice, the dierent nonlinear dependence o on h or bare soil ( b ) and vegetated (v) conditions results in two outcomes: (a) v > b, and (b) the

4 52 S. Thompson et al. / Advances in Water Resources 34 (2) dependence o b on h saturates at much lower values o h than does v Iniltration behavior A recent meta-analysis o biomass-iniltration capacity relationships across nearly 5 vegetation communities [4] indicated that in arid ecosystems, vegetation provides a irst order control on iniltration capacity, with the dependence o iniltration capacity on vegetation describing a power-law relationship with aboveground biomass. Here, the lea area index (LAI) is used as a surrogate or above ground biomass, allowing the iniltration capacity to be expressed as: I c ðtþ ¼að þ klai b ÞGðhÞ ðþ where a indicates the iniltration capacity o bare soil, k a proportional enhancement associated with vegetation, and b.4 based on empirical data [4]. The multiplier G(h) is used to alternate between the depth-dependent iniltration case used in vegetation patterning models, G(h)=h/z o, and a classical description o iniltration where G(h) =. To allow the two cases to be directly compared we choose the baseline iniltration rate a so that I c is equal or the depth-dependent and depth-independent cases when the depth o ponding is equal to the bare-soil momentum roughness height, h = z o. This matching is necessary to ensure that the maximum ponded depths generated in the two cases are comparable. The sorptive eects o the crusted soils were assumed to be small so that iniltration capacity could be assumed to be entirely gravitational. 3. Numerical experiments To address the research questions posed in the introduction, Eqs. () (6) were solved numerically on a lat, D domain, 3 m in length (the scale illustrated in Fig. ). The let hand side o the domain ( 5 m) was treated as a bare soil condition covered by crusted soils, while the right hand side o the domain (5 3 m) was assumed to be covered by uniorm vegetation. The rictional resistance to low (roughness) in bare areas was computed with Eq. (7), assuming z o =. m, i.e. a mm grain roughness, comparable to a lat sandy surace. The roughness in vegetated areas was computed with Eq. (9). No-lux boundary conditions were applied at the lateral boundaries assuming the domain could be viewed as part o a repeating periodic vegetation pattern. We tested the sensitivity o the results to these spatial parameters by varying the proportion o the surace that was vegetated between 25% and 75%, and by shrinking the domain size by an order o magnitude to 3 m. These changes in percentage vegetation cover span the observed variation in the scales o periodic vegetation patterns observed in the ield. Rainall and iniltration properties were chosen to relect realistic but limiting cases:. cm/h iniltration capacity or bare soils (c.. [5]) and storm intensities ranging rom. to 7 cm/h. These intensities span a range rom the lowestintensity storms that could generate ponding given the soil iniltration capacities (comparable to storms with < year return intervals); to intensities comparable to a 5-year storm, assuming storm durations o 3 min (intensity-requency-duration data taken rom cyclonically/monsoonally inluenced desert ecosystems, [4]). For the iniltration cases where G(h)=h/z o, iniltration rates asymptote to zero as the soil surace dries. In these cases the numerical runs were terminated when the ponded volume on the soil surace was <% o the total precipitation volume. Otherwise, simulations ended when the total ponded volume became zero. Parameters or the numerical experiments are summarized in Table Results The results are organized around the two research questions posed in the introduction. Results are presented in terms o cumulative iniltration (F, where Fv and F b are used to represent the cumulative iniltration per unit area or vegetated and bare soil sites, respectively). These values are reerenced to each other and to the total (cumulative) rainall depth per unit area, P tot (mm). Unless otherwise stated, the results assume that turbulent stresses are dominant. 4.. Can contrasts in iniltration rate and surace roughness lead to a lateral redistribution o water rom bare to vegetated sites? All the model runs were compared to a reerence case in which there was no contrast in the iniltration capacity between the bare and vegetated sites (i.e. k = ), and both were treated as the bare case. In this situation, the cumulative iniltration Fv = F b = F independent o any other parameter choice and the D iniltration situation is recovered. However, when a contrast in roughness and iniltration was present, then dierences in the ponded depth between the bare and vegetated sites develop, as illustrated in Fig. 2. These dierences generate lateral low rom bare sites to vegetated sites. The eect o this low is to enhance the water available at the vegetated site beyond that provided by rainall, and to reduce water availability at the bare site to less than that provided by rainall. As illustrated in Fig. 3, this enhancement is dependent upon the storm intensity, the sensitivity o the iniltration behavior to the presence o vegetation (k) and the assumptions made about the iniltration dependence on ponded depth. For h-dependent iniltration (i.e. G(h)=h/z o ), the behavior o this lateral water subsidy provided to the vegetated sites in addition to rainall is consistent, regardless o rainall intensity, at 2% (k =2) or 3% (k = 4). For h-independent iniltration (i.e. G(h) = ), the subsidy is strongly and nonlinearly-dependent on the rainall intensity, and ranges rom to 4% o the rainall depth. The subsidy was most strongly inluenced by the iniltration contrast. Although the roughness contrasts generated during the course o the simulations ranged over an order o magnitude, with b.3 and v 5 as shown in Fig. 2, removing this contrast and setting v = b only increases the cumulative iniltration by a actor o % or the heaviest storms and h-independent iniltration (see Supplementary Fig. ), which is comparable to increasing the iniltration contrast by a actor o 2 (see Fig. 3). Furthermore, this sensitivity to the roughness decreased as the iniltration contrast increased, suggesting that it is most important or low surace gradients. The water contributed to the patch by lateral low does not iniltrate uniormly across the vegetated site but is concentrated at the bare-soil vegetation boundary. As shown in Fig. 4, the proile o the lateral distribution o iniltrated water is invariant in the case o h-dependent iniltration, but varies with intensity where iniltration is independent rom h. In this case, the center o the patch is only subsidized by lateral redistribution or the most intense and inrequent storms. By computing the spatial gradient o the cumulative iniltration volume, a region can be deined where j@f/@xj >, where =. maxj@f/@xj. As illustrated in Fig. 5, panel D, this is the region where the presence o the soil-vegetation contrast appreciably alters the local iniltration due to the action o lateral low. This region can be quantiied by a length-scale L (deined as the length o the region where j@f/@ xj > ), termed the iniltration adjustment length. L deines the component o the bare soil region that contributes water to the vegetated patch via lateral low, and the component o the vegetated patch which receives this redistributed water as iniltration. In the same way that the redistrib-

5 S. Thompson et al. / Advances in Water Resources 34 (2) Fig.. (A) Runon runo dynamics are hypothesized to contribute to the persistence o patchy arid ecosystems such as these ound in South-Eastern Niger, N, E. Image rom Google Earth. Copyright 2 Digital Globe. (B) By assuming that iniltration scales linearly with ponded depth, varies with local vegetation cover, and that low can be approximated by a diusion equation, simple models reproduce the morphology o observed vegetation patterns. The color scale is proportional to local biomass density, normalized between (bare soil) and (complete vegetation cover). Source: Example taken rom [45]. Table 2 Parameters or the model simulations. Where more than one parameter or model component is listed, the cases were run actorially to explore all combinations o parameters. Model component Value(s) Iniltration model G(h)=h/z o or G(h)= Hydraulic model Saint Venant equations, Mannings equation, Nonlinear diusion, Linear diusion Iniltration contrast k (no contrast), 2, 4 Base iniltration rate. cm/h (a, cm/h) Roughness contrast Laminar (no contrast, or 64/Re) or (bare soil to vegetated) Turbulent (no contrast, or Eq. (9)) Rainall intensity (cm/h).,.5, 2.5, 5, 7 cm/h Rainall duration (min) 3 Roughness length (z o, mm) 2 Drag coeicient (C d, ).2 Canopy height (H c, m). (always greater than ponded depth) Lea density [m 2 /m 3 ] uted water is distributed non-uniormly across the vegetated patch, it is apparent that the loss o water to the lateral subsidy occurs non-uniormly across the bare patch, with the subsidy being strongest immediately adjacent to the patch boundary (see Fig. 5). In practice L or the h-dependent iniltration always extends to the domain boundary (Fig. 5 Panel C), implying that all o the bare soil area contributes to the lateral lux, and all o the vegetated patch receives a lateral subsidy. However, the iniltration adjustment length or h-independent iniltration is a unction o the rainall intensity (Fig. 5 Panels A and B). Where L M with M representing the size o the domain, alterations to the spatial coniguration o vegetation and soil, or to the size o the vegetated patch, adjust the importance o the subsidy in trivial ways only. That is, although the ratio o Fv to P tot increases as the size o the vegetated patch shrinks, this simply relects that a relatively larger proportion o the vegetated patch is spanned by L (c.. Fig. 5 panel A). A ixed bare soil area contributes to the lateral luxes, and a ixed vegetated area receives the subsidy, and the Fv to P tot ratio only relects the magnitude o the subsidized area to the whole vegetation patch. However, as P rises, L increases to the point where it spans the domain and interacts with the domain boundary (c.. Fig. 5 panel B). This interaction not only increases the proportion o the vegetated patch spanned by L, but also the magnitude o the subsidy to the patch, due to water backing up rom the no-low condition at the boundary. Thus the boundary condition interaction leads to a greater subsidy eect or a 75% bare area when P = 7 cm/h (Fig. 5 panel B) but not when P =. cm/h (Fig. 5 panel A). The low-boundary interaction is relected in the nonlinear dependence o the subsidy on the patch size or increasing rainall intensities (Table 3). In the case o h- dependent iniltration, the boundary conditions inluence ponded water redistribution or even the lowest intensity storms. However, these eects are invariant with storm size, meaning that the proportional changes in lateral redistribution induced by the patch area are insensitive to the rainall intensity (Table 3). As the rainall intensity increases, the iniltration adjustment length increases in a near linear ashion until L M at which point the boundary conditions interact sensitively with the iniltration behavior (see Fig. 5 panel D). This analysis suggests that the iniltration adjustment length L is a key length-scale dictating the iniltration behavior in the systems where iniltration behaves in an h-independent ashion. The sensitivity o L to P appears to be near-linear. As the iniltration contrast increases, L grows more slowly with P: L / 5P or k =2 but L / 5P or k = 4. The iniltration adjustment length is insensitive to the presence o a roughness contrast in b and v. It is however sensitive to the assumptions about the parameterization o the stresses, and L is constrained at low Reynolds numbers under the assumption that = 64/Re. The emergence o L as a key length scale determining the redistribution o water suggests that adjustments to the spatial scale o the domain M could ensure that all the bare soil area contributes to the lateral low, and that all the vegetated patch beneits rom the redistribution. Assuming L is approximately symmetric across the plant-soil boundary, this optimization would be achieved when M < L, with the L value derived rom the average properties o the storms that generate lateral low at any given location. Based on observations o patchy vegetation, a reasonable lower bound on the patch size is approximately 3 m, compared to the 3 m previously assumed. Re-running the model results suggests that the degree o subsidy or M = 3 or M = 3 is virtually unchanged or low intensity storms. As the storm intensity increases lateral transport or M = 3 is enhanced relative to M = 3, and rainall subsidies increase by approximately 5% (h-independent iniltration). The enhancement o iniltration subsidies was damped or the case o h-dependent iniltration (presumably because the boundary condition inluenced water redistribution in both the 3 and 3 m domains) with a 3% enhancement o iniltration. The results discussed above assume that the two-dimensional orm o the patches can be locally approximated as a linear vege-

6 54 S. Thompson et al. / Advances in Water Resources 34 (2) A Ponded depth (cm) 3.5 h V h/ x B Velocity (cm/s) h / x (cm/m) C 3 6 D.6 Ponded depth (cm) Velocity (cm/s).3.5 h / x (cm/m) 2 3 Distance (m) Distance (m) Fig. 2. Examples o the computed proiles o ponded water depth, riction actors, the surace water slope and the low velocities at the end o a 3 min 7 cm/h storm event or (A) and (B) h-dependent iniltration and (C) and (D) h-independent iniltration. Panels (A) and (C) show the ponded depth (blue, let axis) and the riction actors (black dashed lines, right axis). The green bars indicate the vegetated region. Panels (B) and (D) show the low velocity (with the positive direction being rom let to right) (pink, let axis) and the surace water gradients (light blue dashed lines, right axis). Note that the peak low velocities do not coincide with the maximum surace gradients because o the higher resistance generated by vegetation. (For interpretation o the reerences to colour in this igure legend, the reader is reerred to the web version o this article.) λ=2, h dependent λ=4, h dependent λ=2, h independent λ=4, h independent.4.3 F v / F b F v / P tot P, cm/hr P, cm/hr Fig. 3. Contrast in cumulative iniltration between bare soil and vegetated sites as a unction o the iniltration contrast k, the iniltration behavior (h-dependence or independence) and the rainall intensity. (B) Relative enhancement o vegetation water availability over rainall as a unction o iniltration contrast and iniltration behavior. tated band, a reasonable assumption or labyrinthine or gapped morphologies [8]. For isolated, near-circular patches o vegetation, however, the ratio o the patch area to the length o the patch edge is much smaller. When M = 3, such circular morphology would increase the water availability by 35% or 48% over rainall or an h-dependent iniltration case (or k = 2 and k = 4), or by as much as 62% o rainall depth or h-independent iniltration with P = 7 cm/h and k = 4. The most eicient water transer scenario we considered consists o circular patches, occupying 25% o the domain, with a short wavelength o 3 m. Under these conditions, the total lateral subsidy amounts to an additional % o the rainall received by the patch. This is consistent with the observations o patchy vegetation orming isolated spot patterns in very dry climates [8] How sensitive are the predictions about the lateral redistribution to the assumptions made about iniltration, roughness, and to the choice o hydraulic model used to parameterize the low? As illustrated in Fig. 3, the assumptions about iniltration behavior were the dominant actor in determining the redistribution o water or cases where it is assumed that the stresses are ully turbulent. Changes in iniltration behavior altered the sensitivity o the redistribution to the rainall intensity, and altered the magnitude o the redistribution by actors o up to %. By comparison, doubling the iniltration sensitivity to the presence o vegetation (i.e. altering the iniltration contrast by 7% or the parameters chosen) resulted in only a % change in the volume o water iniltrated over the vegetated patch. I the roughness

7 S. Thompson et al. / Advances in Water Resources 34 (2) cm/hr 5 cm/hr 2.5 cm/hr.5 cm/hr. cm/hr F v (x)/f v (x=3) x(m) x(m) Fig. 4. Normalized cumulative iniltration (F(x) = F(x)/F(x = 3)) over the vegetated sites. (A) For h-dependent iniltration, all curves collapse to a single power law relationship o the orm F(x) = x Some lateral transer occurs at all scales. (B) For h-independent iniltration the lateral extent o the subsidy depends on the intensity o rainall and is constrained to less than the ull width o the vegetated patch or intensities o 2.5 cm/h or less. As rainall intensities increase, a greater proportion o the lateral subsidy extends to the boundary (at x = 3 m), and consequently the iniltration contrast through space- or the magnitude o the F(x) = F(x)/ F(x = 3) line plotted here declines as the intensity increases. P=. cm/hr 25% b 5% b 75 % b 8 P=7 cm/hr h dependent A B C F (cm) x (m) x (m) P =. cm/hr.5 L D 3 E F / x (cm/m).5.5 L (m) 2 L<M L > M x (m) P (cm/hr) o Fig. 5. Sensitivity o the iniltration behavior to the spatial organization o the domain. (A) (B) show the behavior o h-independent iniltration or rainall rates o. cm/h (A), and 7 cm/h (B). The distribution o iniltration through space responds sensitively to the changes in rainall intensity. b indicates the percentage o the domain set to bare soil conditions. (C) shows a single case o h-dependent iniltration (P =. cm/h) which exhibited spatially similar behavior regardless o the rainall intensity. (D) shows the deinition o the adjustment length L, as the region over which j@f/@xj >. (E) indicates the behavior o L as the rainall intensity P is increased. Two regions emerge: one in which L increases in a near-linear ashion and is less than the domain width M. Once L grows to greater than M the perturbed region spans the domain and no longer changes with rainall intensity. contrast was eliminated altogether, the increase in the lateral redistribution o water to the vegetated patches was negligible or h-dependent iniltration and only increased by % or h-independent iniltration (see Supplementary Figs. and 2). These sensitivities were not strongly contingent on the patch sizes. I laminar stresses were included under the assumption that these could be represented as 64/Re, there was no discernible eect on the spatial distribution o cumulative iniltration in the h- dependent case (that is, or k =2,Fv/P tot =.2 or all rainall intensities. There were large eects on the h-independent case however, with the presence o laminar stresses greatly reducing the subsidy o water to the vegetated sites. The maximum values o Fv/F b =.2 in the laminar case compared to.9 in the turbulent case, and the maximum value o Fv/P tot =. or the laminar case compared to.3 in the turbulent case (to compare values or all rainall intensities, see Supplementary Fig. 3). The choice o hydraulic model used to describe the low behavior also introduced dierences in the predictions o water subsidy to the vegetated patch, as shown in Fig. 6. Deviations on the order o 5 2% rom the predictions o the Saint Venant equations were

8 56 S. Thompson et al. / Advances in Water Resources 34 (2) Table 3 Fv/P tot reported or ive dierent rainall intensities and three dierent patch sizes. b indicates the bare soil area, and v the vegetated area. P (cm/h) 25:75 b:v 5:5 b:v 75:25 b:v 5:5 b:v M = 3 m M = 3 m M = 3 m M =3m h-independent iniltration h-dependent iniltration associated with the use o linearized or simpliied low models. These errors are comparable to the magnitude o the enhancement o iniltration in the vegetated sites induced by the soil contrast (c.. Fig. 3). The exception is the linear diusion model in combination with h-dependent iniltration, which reproduces the predictions rom the Saint Venant equations closely. Coincidentally (and ortunately), this is the combination o iniltration and hydraulic assumptions used in the majority o patchy arid ecosystem models [6,8,25,26], suggesting that the choice o a simpliied hydraulic model is unlikely to have signiicantly biased the predictions o these models in situations where h-dependent iniltration occurs. 5. Discussion and conclusion The analysis here suggests several broad conclusions: irstly that on lat sites, the iniltration contrast can induce signiicant lateral transport o water during storm events, secondly that the nature o the iniltration assumptions and iniltration contrast are the most signiicant controls on this transport, with the hydraulic actors such as surace roughness acting as secondary eects, and inally that the redistribution behavior can be approximated by a lengthscale that appears to increase in a near-linear ashion with the rainall intensity. Applying a simple scale analysis to the steady-state Saint Venant momentum equation allows us to briely explore these limitations o validity or this analysis. For instance, rom the results o the simulations shown in Fig. 2, the peak magnitude can be estimated as approximately 5 mm over 5 m, or a.% slope. bed slopes were much greater than.%, the bed slope terms would dominate over the ree surace gradient and a more conventional hydraulic analysis should be applied. Thereore, lat in this context is bounded by slopes on the order o.%, or around.. The model results have several implications or understanding the dynamics o water redistribution in patchy arid ecosystems. Firstly, they suggest that even during intense storms on soils with a strong iniltration contrast, the total volume o water redistributed laterally due to the slope produced by the ree water surace is a relatively small raction o rainall. No simulation suggested more than a % enhancement o water availability (i.e. Fv/ P tot = 2.) in the vegetated sites, and then only or the largest and least requent storms and with bare soil covering 75% o the domain. This is much smaller than the reported enhancements in water stored in the soil proile beneath vegetated sites on slopes (up to eight times the volume o annual rainall as reported by Galle et al. [42] or sloping sites in the Sahel, where redistribution contributed approximately our times more water to the vegetated zone annually than did rainall). The relatively small increases in water availability introduced in this setting suggest that, with the possible exception o sites where vegetation grows in very small, circular patches, it would be sensible to consider that surace redistribution o water operates in conjunction with other acilitative processes to sustain vegetation. This is particularly likely to be true in situations where microtopographic variation exists between bare and vegetated sites, since microtopographic variability is likely to impede lateral redistribution, or at least to conine redistribution strongly to the edges o vegetated mounds. This conclusion broadly supports the likely importance o belowground processes (i.e. redistribution o iniltrated water across the plant root system) and the plant structural parameters or sustaining vegetation growth in lat sites as suggested by Leever et al. [7]. This would also be consistent with the observations o vegetation patterning arising in locations where no iniltration contrast is present [5], and raises the possibility that dierent acilitative processes may sustain vegetation patches in sloping versus topographically lat sites.. h dependent SVE(λ = 2) Manning(λ = 2) Nonlinear(λ = 2) Linear(λ = 2).25 h independent F v /F v (SVE) P, cm/hr P, cm/hr Fig. 6. Cumulative iniltration in the vegetated zone as computed using three simpliications to the Saint Venant equations, relative to the cumulative iniltration computed by the complete Saint Venant equations. Note that where iniltration was proportional to ponded depth, the linear diusion approximation provided an excellent estimate o cumulative iniltration in the vegetated sites, while a rating curve or a nonlinear diusion approximation induced 5 2% error. Where iniltration was independent o depth, the error in each o the models ranges between 5% and 2%.

9 S. Thompson et al. / Advances in Water Resources 34 (2) The 2 6% increases in water availability attributable to runon runo mechanisms in sites with approximately 5% plant cover (c.. Fig. 2), while likely insuicient to ully explain the patchiness o dryland vegetation, would contribute to the survival and growth o vegetation within patches, particularly at the edges. They may, or instance, explain the presence o a zone o annual plant growth oten reported at the edge o vegetated patches [4]. It is also in this zone that the roughness o the vegetated sites has its most pronounced eects, particularly or larger storm events, with increasing roughness tending to isolate the contribution o redistributed water close to the edge o the patch (see Supplementary Fig. 2). The importance o the lateral contribution will depend on the statistics o rainall and the iniltration properties at a site. For instance, taking 2 years o rainall statistics rom Menindee, Australia, where vegetated patches orm on plains with slopes as low as.%, [43,5] and simply weighting the events by requency, we would predict that on average lateral subsidies contribute some additional 26% o rainall or approximately 7 mm/year to the maintenance o these vegetated sites (storms with intensities o over. cm/h contributed 98% o all rainall at this site). Secondly, the results suggest that to draw irm conclusions about the dynamics o water redistribution in topographically lat sites with contrasting soil properties, two aspects o the hydraulics must be better constrained. The irst o these relates to the potential links between the depth o ponded water and the dynamics o iniltration. Classical iniltration theory suggests that the spatial partitioning o rainall would be strongly dependent on storm intensity and the nature o the dominant stresses in the luid (turbulent vs. laminar). I, however, iniltration rates are phenomenologically dependent on water depth, relecting small scale microtopographic variations or surace cracking [29,27], the dynamics become relatively independent o rainall intensity and other hydraulic assumptions. This is because an h-dependence o the iniltration behavior means that a pseudo-steady ponded water condition exists at which the iniltration rate approximately balances the rainall intensity. The implications o such a pseudo-steady state are that (a) water ponds across the entire proile, (b) the total variability in the ponded depth across the range o rainall intensities tested is constrained: a actor o six variation in comparison to a 6-old variation or the h-independent case; and (c) the proportional contrast in ponded depth between the vegetated and unvegetated sites is independent o the rainall intensity (at steady state). This convergence to a constant contrast in ponded depth is the dominant eature o the dynamics o the h-dependent simulations, causing the rainall spatial partitioning to be relatively independent o assumptions about roughness, laminar or turbulent low regime, or the imposed rainall variability. The potential or the h-dependent iniltration dynamics to reach a steady state may also explain why the linear diusion model, which is based on the assumption o steady-state inertial terms, was able to reasonably reproduce the results o the Saint Venant equations. By comparison, the h-independent iniltration is always transient, and oten leads to low ponded depths. These low ponded depths signiicantly slow the lateral water lux due to the eedback between depth and low rate. Consequently the redistribution behavior is much more sensitive to assumptions about roughness and low regime in such a setup, and deviations between the Saint Venant results and those o the simpliied low models were always observed. The sensitivity to the speciication o the roughness and low motivates the need to urther investigate the nature o the stresses opposing lateral low. As identiied above, the assumption that all stresses are laminar results in a very eective inhibition o lateral luxes or all but the largest storms, and signiicantly inhibits the capacity o iniltration contrasts to promote spatial partitioning o iniltration volumes on lat terrain. Ascertaining the appropriate parameterization o laminar stresses with respect to the Reynolds Number is a necessary step to reining estimates o the signiicance o overland low. Considerable uncertainty still prevails in the literature as to the appropriate way to model the stresses in partially inundated shallow low over a rough boundary [34,32,24]. Experimental investigations or this speciic orientation o bare soil and vegetation are likely to be the most appropriate way orward. The modeling study suggests that or the purposes o developing water routing models or patchy arid ecosystems, i iniltration exhibits depth dependence, it may be reasonable to replace the solution o the ull Saint Venant equations by a linearized diusion model [8,26,6,25] without loss o validity o the solution. However, should conventional h-independent iniltration predominate, then neither a rating curve nor diusive approximations to the Saint Venant equations can approximate the water redistribution suiciently well to resolve the behavior predicted by the ull Saint Venant solutions. The results suggest that one control on the redistribution o water is the iniltration adjustment length L, a hydraulic length scale set by soil and rainall properties. Where the characteristic length scale o the vegetation soil spatial patchiness M greatly exceeds L, lateral redistribution o water rom bare sites to vegetated sites is sub-optimal, since there are parts o the domain that are not hydraulically inluenced by the iniltration capacity contrast. This implies that not all bare soil regions eectively contribute water to the vegetated patch, and portions o the vegetated site do not receive water via run-on. Thus L sets a reasonable upper limit on the spacing o vegetation patches in the landscape. In general we would anticipate that observed patch spacing would be smaller than L. Again using Menindee as an example, we would predict rom the rainall regime that L would be 5 m, while typical repeating units in this area are instead ound on scales o 5 m [44]. In closing, the numerical experiments perormed here suggest that there are several mechanistic uncertainties regarding the topographically-lat lateral redistribution problem: the behavior o iniltration and the description o the dominant stresses in the luid. These uncertainties are amenable to experimental investigation, either in ield or controlled laboratory conditions (given their physical nature). Clariying these uncertainties will contribute to understanding the eco-hydrological processes maintaining vegetation growth in patchy arid ecosystems and will acilitate improved eco-hydrological modeling o these systems. Acknowledgements The authors acknowledge support rom the NSF through EAR 3339, the NSF Graduate Fellowship program and the Fulbright Italy distinguished scholars program. Appendix A. Supplementary data Supplementary data associated with this article can be ound, in the online version, at doi:.6/j.advwatres Reerences [] Borgogno F, D Odorico P, Laio F, Ridoli L. Mathematical models o vegetation pattern ormation in ecohydrology. Rev Geophys 29:47. [2] Bromley J, Brouwer J, Barker AP, Gaze SR, Valentin C. The role o surace water redistribution in an area o patterned vegetation in a semi-arid environment, south-west Niger. J Hydrol 997;98( 4): 29. [3] Kei S, Rietkerk M, Alados CL, Pueyo Y, Papanastasis VP, ElAich A, et al. Spatial vegetation patterns and imminent desertiication in Mediterranean arid ecosystems. Nature 27;449(759):U23 5.